Filtration de Johnson et groupe de Torelli modulo p, p premier

Abstract

Let S ( g , 1 ) be a connected, compact, oriented surface of genus g, with one boundary component and Mod ( g , 1 ) its mapping class group. Let p be an integer, either equal to 0, or a prime ⩾2. We construct a central p-filtration of Mod ( g , 1 ) , denoted { M ( k , p ) : k ∈ N ∗ = N − { 0 } } , generalizing the Johnson filtration (which corresponds to p = 0 ) such that M ( 1 , g ) = Mod ( g , 1 ) , M ( k , p ) M ( k + 1 , p ) ( k ⩾ 2 ) is a finite dimensional Z / p Z -vector space and M ( 2 , p ) is the Torelli group ( mod p ) (e.g. the subgroup of Mod ( g , 1 ) of homeomorphisms which induce identity on H 1 ( S ( g , 1 ) ; Z / p Z )). We announce the following results: the Torelli group ( mod p ) is generated by the usual Torelli group and the p-th powers of all Dehn twists. We compute the abelianization of the Torelli group ( mod p ) , up to finite 2-torsion. Any Q -homology sphere Σ 3 is obtained by gluing two handlebodies by an element of the Torelli group ( mod p ) , for any prime p ⩾ 3 dividing ( n − 1 ) , where n is the cardinal of H 1 ( Σ 3 ; Z ) . Finally we propose a conjectural invariant for these Q -homology spheres. To cite this article: B. Perron, C. R. Acad. Sci. Paris, Ser. I 346 (2008).